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Tensor Networks and You Nikko Pomata How do you network tensors? Tensors: a review The tensor-network notation T ENSOR N ETWORKS Tensor network examples and You Tensor network methods Matrix product states (MPS) Projected Entangled Pair
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
TENSOR NETWORKS
Nikko Pomata
Stony Brook University
Grad Talks: April 25 2018
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
OUTLINE
HOW DO YOU NETWORK TENSORS?
Tensors: a review The tensor-network notation Tensor network examples
TENSOR NETWORK METHODS
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
TENSOR NETWORKS AND YOU: FRONTIERS IN TENSOR NETWORK RESEARCH
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
REMINDER: WHAT ARE TENSORS, ANYWAY?
Basic idea A tensor is a linear combination of tensor products of vectors: T =
v(j)
1 ⊗ v(j) 2
⊗ · · · ⊗ v(j)
n
=
Ti1,i2,...,inei1 ⊗ ei2 ⊗ · · · ⊗ ein in terms of basis vectors More formally: A tensor is a multilinear map on vector spaces Ti1,...,imj1,...,jn =
· · ·
T
· · ·
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
REMINDER: WHAT ARE TENSORS, ANYWAY?
A tensor is not
A quantity that transforms covariantly with coordinate
changes (a tensor field)
An observable that has multiple indices which
transform under SO(3) rotations (a tensor operator)
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
TENSOR PRODUCTS IN QUANTUM PHYSICS
If a quantum system is comprised of two subsystems A and B, represented by Hilbert spaces HA and HB, then the overall Hilbert space is HA ⊗ HB These subsystems can be:
The states of two different particles The position of a particle along the x axis, and the
position of the same particle along the y axis
A particle’s position, and the same particle’s spin The number of bosons in each of two different states
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
A MANY-BODY SPIN SYSTEM
Often wind up studying lattice systems where every lattice site is a finite-level system (e.g. the spin of an atom at that site) If there are N sites represented by Cd: then H =
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
DRAWING TENSORS
In tensor-network notation, a tensor is drawn as a shape and its indices are drawn as lines: Aijkl = j i l k vijk = i j k uijij = i j i j
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
PUTTING TOGETHER TENSOR NETWORKS
A tensor network is linked together by contractions (just like in GR)
i j l k m p r q
=
uipjqAkrplvrqm
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
EXAMPLE: THE ISING PARTITION FUNCTION
Z(β) =
exp −β
sisj =
e−βsisj
s s
= e−βss
sa sb sc sd
= δsa,sb,sc,sd
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
EXAMPLE: THE ISING PARTITION FUNCTION
Z(β) =
exp −β
sisj =
e−βsisj
s s
= e−βss
sa sb sc sd
= δsa,sb,sc,sd
sa sb sc sd
= saδsa,sb,sc,sd To obtain smsn:
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
MANIPULATING TENSORS: UNITARIES AND ISOMETRIES
Often wind up dealing with unitary and isometric tensors:
u u†
=
u u†
=
i i j j
= δiiδjj
w w†
= ,
w w†
= P
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
MANIPULATING TENSORS: SVD
The singular value decomposition: A = USV†
A is an arbitrary matrix S is diagonal U and V are unitary
In order to turn a matrix equation into a tensor equation, group indices: A = S U V†
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
HOW DO YOU NETWORK TENSORS?
Tensors: a review The tensor-network notation Tensor network examples
TENSOR NETWORK METHODS
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
TENSOR NETWORKS AND YOU: FRONTIERS IN TENSOR NETWORK RESEARCH
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
OBTAINING THE MATRIX PRODUCT STATE
Start with any state on a spin chain (Cd ⊗N) Apply SVD to the state to divide it into two parts
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
OBTAINING THE MATRIX PRODUCT STATE
Start with any state on a spin chain (Cd ⊗N) Apply SVD to the state to divide it into two parts
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
OBTAINING THE MATRIX PRODUCT STATE
Start with any state on a spin chain (Cd ⊗N) Apply SVD to the state to divide it into two parts ... then keep applying SVD to remaining parts to
divide the state site by site
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
OBTAINING THE MATRIX PRODUCT STATE
A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 S0 S1 S2 S3 S4 S5 S6 S7 S8
Start with any state on a spin chain (Cd ⊗N) Apply SVD to the state to divide it into two parts ... then keep applying SVD to remaining parts to
divide the state site by site
This expresses amplitudes as matrix products:
i0i1 · · · iN−1 |ψ = A(i0) S0A(i1)
1
S1 · · · SN−2A(iN−1)
N−1 However, the bond dimension in the middle is ∼ 2N/2
Definition The dimension of an index that is contracted over to
(χ), as opposed to the physical dimension d.
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
USING MATRIX PRODUCT STATES
Calculating observables: apply “transfer matrices” to
“boundary conditions”
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
USING MATRIX PRODUCT STATES
Calculating observables: apply “transfer matrices” to
“boundary conditions”
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
USING MATRIX PRODUCT STATES
Calculating observables: apply “transfer matrices” to
“boundary conditions”
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
USING MATRIX PRODUCT STATES
Calculating observables: apply “transfer matrices” to
“boundary conditions”
Exponential growth of bond dimension χ is a big
problem: makes this harder than just using the
Solution: Truncate the bond dimension - fix it at an
artificially small value (often ∼ 10 − 100). For the ground state of a “typical” local Hamiltonian, singular values will fall off quickly enough that this is a good approximation.
This is the fundamental approximation of tensor
networks.
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
FINDING MATRIX PRODUCT STATES
Usually we’re trying to find or approximate the ground state of a
Variational
Find a ground state by minimizing
ψ |H| ψ = E ψ |ψ , remove all instances of the tensor Ai and solve the resulting generalized eigenvalue problem. This is the Density Matrix Renormalization Group method, or DMRG, originally formulated by Steven White without matrix product states.
Projection
Take an arbitrary state |ϕ , and estimate e−βH |ϕ , in the limit β → ∞. Use the Suzuki-Trotter expansion to approximate e−βH as a product of “small” two-site operators. Use a truncated SVD to return the state to MPS form after successive
Time-Evolving Block Decimation method, or TEBD. These are the two general approaches we apply to find
tensor-network ground states
These methods are exact, up to machine precision, as χ → ∞. Can apply to infinite, homogeneous spin chains: now find
dominant eigenvector of transfer matrices
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
MATRIX PRODUCT STATES IN HIGHER DIMENSIONS?
First attempt: the snake method This is reliable, but extremely inefficient for wide systems
Tensor Networks and You Nikko Pomata How do you network tensors?
Tensors: a review The tensor-network notation Tensor network examples
Tensor network methods
Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization
Tensor Networks and You
THE PROJECTED ENTANGLED PAIR STATE (PEPS)
More elegant, more homogenous, and truly 2D But it can’t be contracted exactly or efficiently ⇓ It’s hard to optimize efficiently